From exceptional collections to motivic decompositions via noncommutative motives
Name
Marcolli-2015-From exceptional.pdf
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Author(s)
Marcolli, Matilde
Trigo Neri Tabuada, Goncalo Jorge
Date Issued
May 2013
Journal
Journal für die reine und angewandte Mathematik (Crelles Journal)
Publisher
Walter de Gruyter
Citation
Marcolli, Matilde, and Gonçalo Tabuada. “From Exceptional Collections to Motivic Decompositions via Noncommutative Motives.” Journal Für Die Reine Und Angewandte Mathematik (Crelles Journal) 0, no. 0 (January 7, 2013). © De Gruyter 2015
Version
Final published version
Abstract
Making use of noncommutative motives we relate exceptional collections (and more generally semi-orthogonal decompositions) to motivic decompositions. On one hand we prove that the Chow motive M(𝒳)[subscript ℚ] of every smooth and proper Deligne–Mumford stack 𝒳, whose bounded derived category 𝒟[superscript b](𝒳) of coherent schemes admits a full exceptional collection, decomposes into a direct sum of tensor powers of the Lefschetz motive. Examples include projective spaces, quadrics, toric varieties, homogeneous spaces, Fano threefolds, and moduli spaces. On the other hand we prove that if M(𝒳)[subscript ℚ] decomposes into a direct sum of tensor powers of the Lefschetz motive and moreover 𝒟[superscript b](𝒳) admits a semi-orthogonal decomposition, then the noncommutative motive of each one of the pieces of the semi-orthogonal decomposition is a direct sum of ⊗-units. As an application we obtain a simplification of Dubrovin's conjecture.
MIT Department
Massachusetts Institute of Technology. Department of Mathematics
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DOI of Published Version
https://doi.org/10.1515/crelle-2013-0027
